A note on testing the covariance matrix for large dimension

We consider the problem of testing hypotheses regarding the covariance matrix of multivariate normal data, if the sample size s and dimension n satisfy . Recently, several tests have been proposed in the case, where the sample size and dimension are of the same order, that is y[set membership, variant](0,[infinity]). In this paper, we consider the cases y=0 and [infinity]. It is demonstrated that standard techniques are not applicable to deal with these cases. A new technique is introduced, which is of its own interest, and is used to derive the asymptotic distribution of the test statistics in the extreme cases y=0 and [infinity].